M Chizhov, M Eingorn - The matrix of nanoscale emitters the cross effects and the tunneling current - страница 1

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ISSN 1024-588Х. Вісник Львівського ун-ту. Серія фізична. 2011. Випуск 46. С. 163-167 Visnyk of the Lviv University. Series Physics. 2011. Issue 46. P. 163-167

 

УДК 537.213 PACS 41.20.Cv

 

THE MATRIX OF NANOSCALE EMITTERS: THE CROSS EFFECTS AND THE TUNNELING CURRENT

M. Chizhov, M. Eingorn

Odessa National University named after 1.1. Mechnikov Dvoryanskaya str., 2, 65026 Odessa, Ukraine e-mail: maxim.eingorn@gmail.com

Cross effects arising in a matrix of nanoscale emitters are studied. When the tips are situated on the surface of the cathode sufficiently far from each other, one can treat them as independent. However, when the tips are closely set, the computation of the tunnelling current must take into account the cross effects and the field strength on their spikes should be decreased accordingly. In this paper we obtain the corresponding formulae.

Key words:   scanning microscopy, cross effects, tunneling current.

 

At present time in different modern instruments, which principle of operation is based on the phenomenon of cold emission, an idea arises to use several or even many conducting tips instead of one, as well as necessity of consistent correct description of their mutual influence, that is cross effects. Changing the shape and geometric sizes of conducting probes, as well as the distance between them, one can achieve optimal relations when using them simultaneously in the scanning tunneling microscope and other analogous instruments with the purpose of the improvement of their work. Let us note that recent papers in the field of scanning tunneling microscopy are devoted to carbon nanotubes [lj and graphene [2-6], that is one of the most perspective directions of modern physics.

Obviously, if tips are situated on the surface of the cathode sufficiently far from each other (in other words, if the average distance l between them is much larger, then the distance d between electrodes, that is l ^> d), then we can neglect their mutual influence and consider them as independent. At the same time cross effects arise between closely set tips (that is for l <€. d). They lead to decrease of the absolute value E of strength E of the electric field close to a spike of every tip (emitter), therefore the corresponding tunneling current also noticeably decreases.

Emitters have nanoscales, being favorable to considerable increase of the quantity

E

ty, generally speaking, calling of mathematical apparatus of quantum mechanics is

l

© Chizhov M., Eingorn M., 2011
V[1]


1, (1)where a, b, d are some positive constants. Assuming b = a, from (1) we obtain the canonical equation of the hyperboloid of revolution:x[2] + у[3] z[4]


-1. (2)Introducing the polar coordinate p = \Jx[5] + y[6] on the plane xy, from (2) we obtain

2       2                 2 2

(- - - = -1,    - - (- = 1. (3)

a[7]     d[8]            d[9] a[10]

This is the canonical equation of the hyperbola in coordinates (z,p). Its vertex has coordinates z = d, p = 0, and the focus has coordinates z = л/a[11] + d[12] = f,p = 0. Following [8], let us construct a family of confocal hyperbolas:

 

Z_____ p__ = 1 (4)

X[13]    f[14] - X[15]      ' ()

where X is a parameter of the considered family. When X = d we obtain the equation

X = 0   z = 0
in this limiting case tips are not independent and cross effects are so perceptible that

E

U/d

Therefore, the proposed formula (7) demonstrates correct asymptotic behavior for very rare and very thick positions of tips on the surface of the cathode. Let us note that it is in good agreement with experimental data and allows determining tunneling current density, and the approach itself, leading to it, can find application in the field of scanning tunneling microscopy.

Obtained in this paper results allow to draw the following conclusion: we have proposed and substantiated the semiempirical formula (7) for the absolute value of strength of the electric field on the spike of one tip in presence of neighboring tips, taking into account cross effects between them and demonstrating correct asymptotic behavior.


1.  Nemes-Incze P. Mapping of functionalized regions on carbon nanotubes by scanni­ng tunneling microscopy / P. Nemes-Incze, Z.Konya, I. Kiricsi et al. //J. Phys. Chem. C. - 2011. - Vol. 115. - P. 3229-3235.

2.  Choudhary Sh. K. Scanning tunneling microscopy and spectroscopy study of charge inhomogeneities in bilayer graphene / Sh. K. Choudhary, A.K.Gupta // Solid State Communic. - Vol. 151, № 5. - P. 396-399.

3.  Yamamoto M. Structure and properties of chemically prepared nanographene islands characterized by scanning tunneling microscopy / M. Yamamoto, S. Obata, K. Saiki et al. // Surf. Interface Anal. - 2010. - Vol. 42. - P. 1637-1641.

4.  Geringer V. Electrical transport and low-temperature scanning tunneling mi­croscopy of microsoldered graphene / V. Geringer, D. Subramaniam, A. K. Michel et al. // Appl. Phys. Lett. - 2010. - Vol. 96. - P. 082114-082116.

5.  Hiebel F. Atomic and electronic structure of monolayer graphene on 6H - SiC(000 - 1)(3 x 3): a scanning tunneling microscopy study / F. Hiebel, P. Mallet, L. Magaud et al. // Phys. Rev. В - 2009. - Vol. 80. - P. 235429­235437.

6.  Peres N. M. R. Scanning tunneling microscopy currents on locally disordered graphene / N. M. R. Peres, Sh.-W. Tsai, J. E. Santos et al.// Phys. Rev. В - 2009.

 

-  Vol. 79. - P. 155442-155451.

7.  Chizhov M. V. The electrostatic field in the two-dimensional region between uneven electrodes / M. V. Chizhov, M. V. Eingorn // Accepted in Photoelectroni-cs. - 2011.

8.  Vereschagin I. P. Foundations of electro-gas-dynamics of dispersed systems / I. P. Vereschagin, V. I. Levitov, G. Z. Mirzabekjan et al. - Moscow : Energy, 1974.

-  480 c.


ІббМАТРИЦЯ НАНОРОЗМІРНИХ ЕМІТЕРІВ: ПЕРЕХРЕСНІ ЕФЕКТИ І ТУНЕЛЬНИЙ СТРУМ

М. Чижов, М. Ейнгорн

Одеський національний університет імені 1.1. Мечникова вул. Дворянська, 2, 65026 Одеса, Україна e-mail: maxim, eingorn @gmail. com

Досліджено перехресні ефекти, що виникають у матриці нанорозмірних емітерів. Якщо голки розташовані на поверхні катоду достатньо далеко одна від одної, ми можемо вважати їх незалежними. Проте якщо голки є розташовані близько, для розрахунків тунельного струму потрібно брати до уваги перехресні ефекти та відповідно зменшити напруженість поля на їхніх вістрях. У цій статті отримано відповідні формули.

Ключові слова: сканувальна мікроскопія, перехресні ефекти, тунельний струм.

 

 

МАТРИЦА НАНОРАЗМЕРНЫХ ЭМИТТЕРОВ: ПЕРЕКРЕСТНЫЕ ЭФФЕКТЫ И ТУННЕЛЬНЫЙ ТОК

М. Чижов, М. Ейнгорн

Одесский национальный университет им. И. И. Мечникова ул. Дворянская, 2, 65026 Одесса, Украина e-mail: maxim, eingorn @gmail. com

Исследованы перекрестные эффекты, возникающие в матрице наноразмер-ных эмиттеров. Если иглы расположены на поверхности катода достато­чно далеко друг от друга, мы можем считать их независимыми. Однако если иглы близко расположены, для расчета туннельного тока следует при­нимать во внимание перекрестные эффекты и соответствующим образом уменьшить напряженность поля на их остриях. В данной статье получены соответствующие формулы.

Ключевые слова: сканирующая микроскопия, перекрестные эффекты, туннельный ток.

 

Статтю отримано: 30.05.2011 Прийнято до друку: 14.07.2011


surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

surface of the anode, and when 0 < A < d we obtain the equation of the some intermediate surface.

Obviously, the potential <p of the electric field in the considered case depends only on the introduced parameter A. Let us write down Laplace's equation A<p = 0, which it satisfies, in cylindrical coordinates, taking into account the cylindrical symmetry of the posed problem:

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M Chizhov, M Eingorn - The matrix of nanoscale emitters the cross effects and the tunneling current